The base of S is the triangular region with vertices (0, 0), (2, 0), and (0, 2). Crosssections perpendicular to the yaxis are equilateral triangles.
Find the volume V of this solid.
3 Answers

Base of solid is region bound by line x+y=2, xaxis and yaxis.
Since crosssections are perpendicular to yaxis, we will integrate with respect to y.
Function: x = 2 – y, limits: y=0 to y=2
Each crosssection is equilateral triangle with base = 2y
Area of each crosssection = 1/2 * (2y) * (2y) * sin(60) = √(3)/4 (2y)²
Integrating from 0 to 2, we get
V = √(3)/4 ∫₀² (2y)² dy
V = √(3)/4 ∫₀² (4 – 4y + y²) dy
V = √(3)/4 (4y – 2y² + y³/3) ₀²
V = √(3)/4 [(8 – 8 + 8/3) – 0]
V = √(3)/4 * 8/3
V = 2√(3)/3
V = 1.1547

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